DISCOVERY OF GOLDEN SECTION IN PARTHENON
By Prof. L. Kaliambos (Kaliambos-Natural Philosophy) January 22 , 2016 NEW REVOLUTION IN METHODS OF ARCHAEOLOGY LED TO MY DISCOVERY OF HEPHAESTION TOMB AND TO MY DISCOVERY OF GOLDEN SECTION IN PARTHENON AND IN CHEOPS PYRAMID This Photo is from the intrview I gave to the author of Spiritual Thasally, Dimitra Bardani, through the TV Thessaly ( Greece) about my discovery of the math of the golden section in Parthenon. '(ΑΝΑΚΑΛΥΨΗ ΤΗΣ ΧΡΥΣΗΣ ΤΟΜΗΣ ΣΤΟΝ ΠΑΡΘΕΝΩΝΑ). '''A new combinatory method for revealing the math in the construction of Amphipolis tomb and of Parthenon led to my discovery of Hephaestion tomb (Nov. 21, 2014) and of the golden section in Caryatids of Erechtheion. Note that such a combinatory method was used not only by the British architect Ventris (who in 1952 deciphered Linear B) but also by Newton who made spectacular discoveries in mathematics (differential calculus), in optics ( particles of light having mass), and in gravity involving 3 laws of motion. Whereas Einstein’s theories of massless quanta of fields did much to retard the progress of fundamental physics. (See my “Newton invalidates Einstein”). ' The Parthenon a marble temple (5th century BC) dedicated to Athena, dominates the Acropolis in Athens. Some studies of the Acropolis, including the Parthenon, concluded that many of its proportions approximate the golden ratio. The Parthenon's facade as well as elements of its facade and elsewhere could be circumscribed by golden rectangles. Moreover my discovery of the obvious golden section in Caryatids of Erechtheion shows that the Greeks have based the design of the Parthenon on this proportion. Phidias (500 BC - 432 BC) and Mnecikles two Greek sculptors and mathematicians, studied the golden ratio and applied it to the design of sculptures for the Parthenon. Plato (circa 428 BC – 347 BC), in his views on natural science and cosmology presented in his “Timaeus,” considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos. In nature this golden section as a principle may be observed in the arrangements of leaves on a twig, petals on a flower and the arms of the starfish. The ancient Greeks considered a rectangle whose sides are in this ratio to be aesthetically the most pleasing of all rectangles and constructed their buildings on this principle. Another ratio does appear throughout most of the Parthenon. For example, the Parthenon is 30.88 meters wide and 69.5 meters long. This equals a 4:9 ratio which cannot be related to the so-called Golden Section. In fact it based on three Pythagorian rectangles. That is, three Pythagorian rectangles of 3 and 4 giving the diagonal of 5 have a total ratio 9/4 = (3/4 + 3/4 +3/4) . Therefore, so far, in the absence of any discovery about the obvious golden section of sculptures this view that the golden ratio was employed in the design of Parthenon had been disputed under the assumption that the Greek math like the golden section with the symbol Φ was first documented in the written historical record by Euclid in “Elements.” (300 BC). It states, “a straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.” In fact, in my paper “Discovery of π and Φ in Giza great pyramid” I showed that both the Pi = π = C/d =3.1416 and the Phi = Φ = (1+50.5)/2 = 1.618034 were used in the construction of Cheops pyramid (2560 BC). Also in my paper “Relation of Pi to Phi and mystic numbers” I showed that such mystic numbers like Φ, 3, π, and 12 were used in the construction of the Mathematical tomb of Hero Hephaestion (320 BC). Golden Section Phi = Φ = (1 + 50.5)/2 is obtained by dividing a line into two parts (α and β) such that the square of the first part is equal to the product of the whole segment (α+β) and the second part. That is α2 = (α +β )β or (α +β )/α = α/β = Φ One method for finding the value of Φ is to write α = Φ and β = 1 (unit length). Therefore (Φ+1)/Φ = Φ/1 or Φ + 1 = Φ2 which can be rearranged to Φ2 - Φ -1 = 0. Today it is well known that in a quadratic formula of the following general form aΦ2 + bΦ + c = 0 the value (positive value) of Φ is given by Φ = [ -b + (b2 -4ac)0.5] / 2a Since a = 1, b = -1, and c = -1 one gets Φ = (1+ (1+4)0.5] / 2 = (1+50.5) / 2 = 1.6180339887…. The Egyptians used the Phi = Φ = 1.6180339887…) in the design of the Great Pyramids and they thought that the golden ratio was sacred. Therefore, they used the golden ratio when building temples and places for the dead. The Egyptians were aware that they were using the golden ratio Φ. However, one should ask how the ancient Egyptians were able to find the solution of the quadratic formula Φ2 - Φ -1 = 0 . The study of this called algebra goes to the antiquity. Recent discoveries have shown that Babylonians and Egyptians solved problems in algebra , although they had no symbols for variables. They used only words to indicate such numbers, and for that reason their algebra has been referred to as theoretical algebra. The Ahmes Papyrus, an Egyptian scroll going back to 1600 BC has a number of problems in algebra, in which the unknown is referred to as a hau, meaning “a heap”. Also practically the so -called Pythagorean theorem (6thcentury BC) was well known to Babylonians and Egyptians. Thus writing 1 + Φ = Φ2 as (1)2 + (Φ0.5)2 = Φ2 one sees that the 1 (unit length) should be the radius r of a cone pyramid, while Φ0.5 = h (height) and Φ = L (slant height). In this case the circumference C =2π because r = 1. In other words such a cone pyramid was believed to be a sacred pyramid because it includes the mystic numbers Phi = Φ and Pi = π . Also the great pyramid of Cheops was believed to be a sacred square pyramid, because a theoretical cone pyramid was inscribed in the square pyramid. '''DISCOVERY OF GOLDEN SECTION IN CARYATIDS OF HEPHAESTION MATHEMATICAL TOMB In my discovery of golden section in Hephaestion tomb I showed that Dinocrates also used the golden section in Amphipolis. The excavation team had so far unearthed two female statues of the Caryatid type in the antechamber, which support the entrance to the second compartment of the tomb. The height α of each Caryatid is α = 2.27 m. The Caryatids are on pedestals of height β = 1.40 m , making the total height (α + β) = 3.67 m of the statues. An approximate value for the Φ is 1.618. Thus Dinocrates starting from the total height (α+β) = 3.67 m was able to determine the heights α and β as (α +β)/α = Φ Or α = 3.67/1.618 = 2.268 m = 2.27 m and β = 3.67- 2.27 = 1.4 m However in the absence of a detailed knowledge about the math and the architecture of ancient Greeks the excavation team in Amphipolis did not relate such very important numbers with the so-called GOLDEN SECTION. DISCOVERY OF GOLDEN SECTION IN CARYATIDS OF ERECHTHEION Erechtheion as seen today was built between 421 and 406 BCE. Its architect may have been Mnesicles, and it derived its name from a shrine dedicated to the legendary Greek hero Erichthonius. Erechtheion, named after the demi-god Erechtheus, the mythical Athenian king, was conceived as a suitable structure to house the ancient wooden cult statue of Athena, which maintained its religious significance despite the arrival of the gigantic chryselephantine statue within the nearby Parthenon. The building also had other functions, though, notably as the shrine centre for other more ancient cults: to Erechtheus, his brother Boutes - the Ploughman, Pandrosos, the mythical first Athenian king Kekrops (or Cecrops) - half-man, half-snake, and the gods Hephaistos and Poseidon. As with the other new buildings on the acropolis, the Erechtheion was built from Pentelic marble which came from the nearby Mt. Pentelicus and was celebrated for its pure white appearance and fine grain. It also contains traces of iron which over time have oxidised, giving the marble a soft honey colour, a quality particularly evident at sunrise and sunset. On the north side, there is a large porch with six Ionic columns, and on the south, the famous "Porch of the Maidens", with six draped female figures (caryatids) as supporting columns. The porch was built to conceal the giant 15-ft beam needed to support the southwest corner over the Kekropion, after the building was drastically reduced in size and budget following the onset of the Peloponnesian war. The best-known and most-copied examples are those of the six figures of the Caryatid Porch. One of those original six figures, removed by Lord Elgin in the early 19th century, is now in the British Museum in London. The Acropolis Museum holds the other five figures, which are replaced onsite by replicas. The five originals that are in Athens are now being exhibited in the new Acropolis Museum, on a special balcony that allows visitors to view them from all sides. The pedestal for the Caryatid removed to London remains empty. From 2011 to 2015, they were cleaned by a specially constructed laser beam, which removed accumulated soot and grime without harming the marble's patina. Each Caryatid was cleaned in place, with a television circuit relaying the spectacle live to museum visitors. Although of the same height = 2.31 meters and build, and similarly attired and coiffed, the six Caryatids are not the same: their faces, stance, draping, and hair are carved separately; the three on the left stand on their right foot, while the three on the right stand on their left foot. Their bulky, intricately arranged hairstyles serve the crucial purpose of providing static support to their necks, which would otherwise be the thinnest and structurally weakest part. Using the same method as that of the two caryatids in the Hephaestion tomb I discovered that the height α = 2.31 m of each Caryatid in Erechtheion ( see the “Caryatids in Athens and Amphipolis”) compared with the height β = 1.43 m of each pedestal gives the ratio Φ = (1+50.5) /2 = 1.618034. That is (α +β)/α = Φ = α/β Or (2.31 + 1.43) / 2.31 = (1+ 50.5)/2 = 2.31/1.4 3 For understanding better such a ratio between the height of Caryatids and the height of pedestals you can see them in the image “Erechtheion from the south” in the “Erechtheion-Wikipedia”. Also fo the golden section based on golden rectangles in Parthenon you can see "The Parthenon and Phi the Golden Ratio". Category:Fundamental physics concepts